Routh-Hurwitz criterion

syms a b c d e f
% Define the coefficients of the polynomial
coefficients = [1 a b c d e f]; 
% Create the Routh-Hurwitz array
RH_array = sym(zeros(7,length(coefficients)));
RH_array(1,:) = coefficients;
RH_array(2,1) = coefficients(1);
RH_array(2,2) = coefficients(3);
RH_array(2,3) = coefficients(5);
for i=3:7
    % Compute the remaining entries in the Routh-Hurwitz array
    RH_array(i,1) = simplify(-det([RH_array(i-2,1) RH_array(i-2,2); RH_array(i-1,1) RH_array(i-1,2)]) / RH_array(i-2,1));
    for j=2:length(coefficients)-1
        RH_array(i,j) = simplify(-det([RH_array(i-2,j-1) RH_array(i-2,j); RH_array(i-1,j-1) RH_array(i-1,j)]) / RH_array(i-2,j-1));
    end
end
% Check the stability criteria using the Routh-Hurwitz array
stable = true;
for i=1:size(RH_array,1)
    if any(RH_array(i,:) == 0)
        stable = false;
        break;
    end
end
if stable
    disp('The polynomial is stable according to the Routh-Hurwitz criterion');
else
    disp('The polynomial is unstable according to the Routh-Hurwitz criterion');
end

The polynomial is unstable according to the Routh-Hurwitz criterion

The results only shows that "The polynomial is instable according to the Routh-Hurwitz criterion". please I want to display the Roots of the polynomial and then submatrix generated from RRouth-Hurwit.

[Sam Chak]

@Sunday, Your Routh-Hurwitz array is incorrectly tabulated.

num = 1;
den = [1 6 15 20 15 6 1];
G   = tf(num, den)

G =
 
                         1
  ------------------------------------------------
  s^6 + 6 s^5 + 15 s^4 + 20 s^3 + 15 s^2 + 6 s + 1
 
Continuous-time transfer function.

ToF = isstable(G)

ToF = logical
   1

% Define the coefficients of the polynomial
coefficients = den

coefficients = 1x7

1 6 15 20 15 6 1

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% Create the Routh-Hurwitz array
RH_array = sym(zeros(7,length(coefficients)));
RH_array(1,:) = coefficients;
RH_array(2,1) = coefficients(1);
RH_array(2,2) = coefficients(3);
RH_array(2,3) = coefficients(5);
for i=3:7
    % Compute the remaining entries in the Routh-Hurwitz array
    RH_array(i,1) = simplify(-det([RH_array(i-2,1) RH_array(i-2,2); RH_array(i-1,1) RH_array(i-1,2)]) / RH_array(i-2,1));
    for j=2:length(coefficients)-1
        RH_array(i,j) = simplify(-det([RH_array(i-2,j-1) RH_array(i-2,j); RH_array(i-1,j-1) RH_array(i-1,j)]) / RH_array(i-2,j-1));
    end
end
disp(RH_array)

% Check the stability criteria using the Routh-Hurwitz array
stable = true;
for i=1:size(RH_array,1)
    if any(RH_array(i,:) == 0)
        stable = false;
        break;
    end
end
if stable
    disp('The polynomial is stable according to the Routh-Hurwitz criterion');
else
    disp('The polynomial is unstable according to the Routh-Hurwitz criterion');
end

The polynomial is unstable according to the Routh-Hurwitz criterion

Answer 1

[Torsten]

Roots of a general polynomial of degree 6 cannot be explicitly computed. You must give numerical values to a,b,c,d,e and f and use "root" or "roots" to get numerical values for the roots.